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The computation of resonances in open systems using a perfectly matched layer


Authors: Seungil Kim and Joseph E. Pasciak
Journal: Math. Comp. 78 (2009), 1375-1398
MSC (2000): Primary 65N30, 78M10
DOI: https://doi.org/10.1090/S0025-5718-09-02227-3
Published electronically: February 6, 2009
MathSciNet review: 2501055
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Abstract: In this paper, we consider the problem of computing resonances in open systems. We first characterize resonances in terms of (improper) eigenfunctions of the Helmholtz operator on an unbounded domain. The perfectly matched layer (PML) technique has been successfully applied to the computation of scattering problems. We shall see that the application of PML converts the resonance problem to a standard eigenvalue problem (still on an infinite domain). This new eigenvalue problem involves an operator which resembles the original Helmholtz equation transformed by a complex shift in the coordinate system. Our goal will be to approximate the shifted operator first by replacing the infinite domain by a finite (computational) domain with a convenient boundary condition and second by applying finite elements on the computational domain. We shall prove that the first of these steps leads to eigenvalue convergence (to the desired resonance values) which is free from spurious computational eigenvalues provided that the size of computational domain is sufficiently large. The analysis of the second step is classical. Finally, we illustrate the behavior of the method applied to numerical experiments in one and two spatial dimensions.


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Additional Information

Seungil Kim
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: sgkim@math.tamu.edu

Joseph E. Pasciak
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: pasciak@math.tamu.edu

DOI: https://doi.org/10.1090/S0025-5718-09-02227-3
Keywords: Perfectly matched layer, PML, resonances, nonsymmetric eigenvalue problem
Received by editor(s): July 9, 2007
Received by editor(s) in revised form: July 22, 2008
Published electronically: February 6, 2009
Additional Notes: This work was supported in part by the National Science Foundation through grant DMS-0609544.
Article copyright: © Copyright 2009 American Mathematical Society